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| Natura: | Preprint |
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2017
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| Accesso online: | https://arxiv.org/abs/1711.06596 |
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| _version_ | 1866917808840900608 |
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| author | Alonso, Ricardo Gamba, Irene M. Tasković, Maja |
| author_facet | Alonso, Ricardo Gamba, Irene M. Tasković, Maja |
| contents | We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to thermodynamical equilibrium and propagation of singularities and regularization of such solutions. All these results are valid under the mere Grad's cut-off condition for the angular scattering kernel. Highlights of this contribution include: (1) full range of $L^{p}$-norms with $p\in[1,\infty]$, (2) analysis for the critical case of Maxwell interactions, (3) propagation of fractional Sobolev exponential tails using pointwise conmutators, and (4) time asymptotic and propagation of regularity and singularities under general physical data. In many ways, this work is an improvement and an extension of several classical works in the area; we use known techniques and introduce new and flexible ideas that achieve the proofs in an elementary manner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1711_06596 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation Alonso, Ricardo Gamba, Irene M. Tasković, Maja Mathematical Physics 82B40, 45Gxx We present in this document the Lebesgue and Sobolev propagation of exponential tails for solutions of the homogeneous Boltzmann equation for hard and Maxwell interactions. In addition, we show the $L^{p}$-integrability creation of such tails in the case of hard interactions. The document also presents a result on exponentially-fast convergence to thermodynamical equilibrium and propagation of singularities and regularization of such solutions. All these results are valid under the mere Grad's cut-off condition for the angular scattering kernel. Highlights of this contribution include: (1) full range of $L^{p}$-norms with $p\in[1,\infty]$, (2) analysis for the critical case of Maxwell interactions, (3) propagation of fractional Sobolev exponential tails using pointwise conmutators, and (4) time asymptotic and propagation of regularity and singularities under general physical data. In many ways, this work is an improvement and an extension of several classical works in the area; we use known techniques and introduce new and flexible ideas that achieve the proofs in an elementary manner. |
| title | Exponentially-tailed regularity and time asymptotic for the homogeneous Boltzmann equation |
| topic | Mathematical Physics 82B40, 45Gxx |
| url | https://arxiv.org/abs/1711.06596 |